Local gradient estimate on Riemannian manifold with asymptotically non-negative Ricci curvature

Abstract

In this paper, we study the asymptotic behavior of the energy densities of harmonic maps, exponentially harmonic functions and positive $p$-harmonic functions at infinity of a Riemannian manifold with asymptotically non-negative curvature. We prove that the energy densities of bounded harmonic maps, exponentially harmonic functions and positive $p$-harmonic functions all vanish at infinity.